Abstract
In this paper, we study effect algebra-induced partial ordered sets. All possible cases of effect algebras generated by bounded partial ordered set of height 2 are given. In addition, the structure of chain effect algebra is studied carefully and the corresponding results are obtained.
1. Introduction
In the past few decades, algebraic structure models used to describe objective things have emerged in large numbers, providing effective tools for our scientific research. In the field of quantum mechanics and quantum logic, there are effects algebra [1], mv-algebra and bck-algebra [2]. In particular, effect algebra greatly promotes the rapid development of quantum theory and quantum logic.
Effect algebra is an important concept introduced by Foulis and Bennett through algebraic abstraction when they studied quantum logic. Since 1994, the study of effect algebra has been favored by scholars.First, Foulis and Bennett gave a series of basic properties of effect algebra [1]. In 1996, Gudder proposed the concept of accurately measurable elements and pivot elements [3], and proved that the effect algebra in which all elements are pivot elements is an orthogonal modular lattice. In 1999, Riecanova removed the condition of existence of identity elements in the effect algebra and obtained the generalized effect algebra [4]. In 2002, Gudder and Greechie extracted some properties of sequence product operation in Hilbert space and proposed sequence effect algebra [5]. In 2019, Wu et al. proposed L-algebras in [6], which is a generalization of homogeneous effect algebra. The relationship between effect algebras and other algebraic structures has also been studied extensively.
Generally speaking, effects algebra is an algebraic structure with a binary partial operation and a unary operation, and contains elements 0,1. The elements of effects algebra are events that are not sharp or clear, such as fuzzy events, quantum effects. Therefore, we can think of effect algebra as fuzzy and ambiguous quantum logic [7]. Effects algebra, of course, is also an algebraic abstraction of the various physical models of quantum mechanics. At the same time, effect algebra is in the category of partial order structure, it is MV-algebra.
As we all know, effect algebra has been applied to quantum theory and quantum logic with great success. However, because effect algebra is a defect of partial algebra, its algebraic structure is not perfect, which brings inconvenience to the application research. The partial binary operation + of effect algebra are fully embodied in the partial order relation introduced. Therefore, it is a good idea to study effect algebra from the perspective of partial ordered sets.
To avoid the difficulty of partial algebras, starting from the concept of effect algebras, we use effect algebras to induce the unique partial order and make it a partial order set. We study effect algebras from this perspective. The types, quantities and structures of partial ordered sets are discussed in effect algebras. The results show that this method can give some interesting theorems describing the structure of effect algebras perfectly.
Below, we first give the basic definition and some properties used in the paper.
2. Preliminaries
First we introduce the concept of effects algebra and some properties that will be used.
Definition 1. [1] An effect algebra ( EA for short) is a systemwhereand is a partial binary operation onsatisfying the conditions: (E1) for all , is defined is defined, and ; (E2) for all is defined is defined, and ; (E3) for all there is a unique such that ; (E4) for all , if is defined then .
The mapping is a total unary operation on We can define the so-called induced order on by stipulation
Since is true for any , is a bounded poset, denoted by . When is a lattice-order relation, we call the effect algebra a lattice-order effect algebra (LEA for short). When is a total-order relation, we call the effect algebra a chain effect algebra (CEA for short).
Definition 2. Two effect algebras and are isomorphic if there is a bijection from to such that for every in the following four equations hold: (I1) is defined is defined and then ; (I2) ; (I3) ; (I4) .Such a is called an isomorphism , denoted by for short).
Definition 3. (see[8]). Letbe a poset, and.(1)coversin, denoted by, ifand, In a poset with a smallest element 0, the element covering the 0 is called an atom. Let is an atom }.(2) are called comparable ifor. Otherwiseandare incomparable , which denoted by.
The following are the basic properties of effect algebra given by Foulis and Bennett in 1994, which we will use.
Lemma 1 (see[1]). Let be an EA and . Then(1) is defined ;(2)if is defined then is defined for all and ;(3) ;(4) ;(5)if and is defined then is defined and ;(6).
Theorem 1. The necessary and sufficient condition for the EA and to be isomorphic is that there exists a bijective from to such that the following condition (I1) is true. (I1) is defined is defined and ,
Proof. : Since is a bijection, then , such that . HenceBy (E3), we have . Therefore, , and , which shows that (I2) and (I3) holds.
For every , Since , we haveHence , i.e. (I4) holds. Therefore, . : By Definition 2, it is trivial.
Definition 4. By an isomorphism between two posets and , is meant a one-one correspondence between and such thatTwo posets are called isomorphic iff there exists an isomorphism between them, we writeor just; an isomorphism of a partly ordered set with itself is called an automorphism. A many-one corregiondence satisfying (4) is called isotone .
By the converse of a relation is meant the relation such that if and only if .
Definition 5. By the dual of a poset is meant that poset defined by the converse relation on the same elements.
According to Definitions 2 and 4, we can obtain the following lemma. It gives the conclusion that the isomorphism of EA isomorphism can imply the order isomorphism.
Lemma 2. Let and be EAs, . Then .
The condition (I1) can be further simplified by the following theorem.
Theorem 2. Two effect algebras and are isomorphic iff , and for all (I1) if is defined then is defined and , (li)
Proof. : First of all, since , is bijective.
For all , is defined, then so does andThus .
Since is a bijection, then , such that . HenceBy (E3), we have . Thus and .Therefore, which shows that (I2) and (I3) holds.
For every , Since , we haveHence , i.e. (I4) holds.
The following is the proof of condition (I1)′ and (li) implication condition (I1).
Let and is defined, in . Thentherefore, such that . By (I1), we have . Then and . i.e. (I1) holds.
Therefore, .
:By Definition 2 and Lemma 2, it is trivial.
Remark 1. The condition (I1)′ in Theorem 1 does not imply the condition (I1), see Example 1.
Example 1. Let . It is easy to verify that and are effect algebras, where , see the follows.
Let . Then is a bijection from to , and satisfies condition (I1)′. But the effect algebras and are not isomorphic.
For each effect algebra , a unique partially ordered set can be obtained under (1). Then, for a given bounded poset , can we introduce partial binary operation and unary operation′such that is an EA and the induced order relation on is exactly ?
The following counterexamples answers this question.
Example 2. Let , (see Figure 1).
If is an EA. Since , we have . Similarly, since , we have , hence . This is a contradiction. Thus, cannot be constructed as an EA.
Example 3. Letand partial binary operations , and unary are defined by
It is easy to prove that and are two completely different effect algebras, but they both induce partial ordered sets of (see Figure 2). In face, We have more general examples.
Example 4. Let is a set of real numbers) and defineandas follows:
Then is a LEA with the induced order . For all , since
Thus, if , then , i.e. , where is usual orders on . Hence .
Conversely, let , i.e. , then we have and , , hence , i.e. . Thus .
Theorem 3. Let be a bounded poset with ( ). If , then poset can be used to construct an EA.
Proof. (1)When , the statement is clearly true.(2)If , then , , , and is an EA.(3)Since is a bounded poset, when , we have , and , i.e. poset is a 3-element chain. Define and′ as follows: therefor, is an EA.(4)For a bounded poset , there are two cases when 4, one is a chain of four elements and the other is (see Example 1). So by Theorem 2 and Example 1, we get , which can be converted into EA. The proof is complete.
Remark 2. These results can be summarized in the following table.
We use to denote -element effect algebras .
Remark 3. Example 2 shows lattice with the least number of elements in non-effect algebra.
Lemma 3. Let be an EA. Then, that is. is automorphic.
Proof. for all , if , then by Lemma 1 (3). And if , thenby Lemma 1 (4). Thus .
Theorem 4. In the isomorphism sense, there are only four types of effect algebra for five elements, which are:(1). Define and as follows:(2). Define and as follows:(3). Define and as follows:(4). Define and as follows:
Proof. Since there are only three kinds of automorphic bounded five-element partial ordered sets: , and , the theorem holds.
Remark 4. The five-element effect algebras whose induced poset is are not unique. There are altogether four of them. We’ll look at this in the next section.
Corollary 1. Let be an EA with ( ). If , then is a LEA.
Example 5. [6] Theis an EA, wheresee the follows.
Let be a poset, and . Here are the definitions of the intervals:
Note that is not a lattice, next, we will consider lattice-ordered effect algebras.
3. Homo-Ordered Effect Algebras
Next we will study the poset induced by the effect algebra with the same property, and first give the definition of the same order effect algebra.
Definition 6. Two effect algebras and are called Homo-ordered if the posets and are isomorphic, denoted by .
In Example 3, and in Example 4, holds. Below, we have more general results.
Theorem 5. Let be an EA, be a poset. If, then , where , :
Proof. First, we prove that is an EA.
For all , if is defined, then is defined, and is defined, hence is defined andi.e. (E1) holds. Similarly, we can prove that (E2) holds as well.Then (E3) holds.
Let is defined , then is defined. Thus, ,i.e. (E4) holds. Hence is an EA.
Next, we show that . i.e. .
Let and . Then we have and , .i.e.Therefore , i.e. .
Since , then , . therefore we have . Hence . Since , we have . Thus, we conclude that holds as well. Therefore, and are isomorphic.
Thus, , the proof is complete.
Remark 5. (1)This theorem gives a way to construct a new EA from the poset of an EA.(2)This method is not sufficient, see Example 3, holds, butanddo not satisfy the relationship of Theorem 5.
Definition 7. Let and are effect algebras and . If we putfor all then is EA, we call a union effect algebra of and , denoted by (see Figure 3.
In Example 3, if we put , then .
Definition 8. Let and are EA. If we putfor all , obviously is EA, we call a direct product effect algebra ofand.
In Example 3, if we put and , then .
If all sub-chains in a poset contain at most element , then we say that the height of the poset is , denoted by .
Lemma 4. Let be a bounded poset with , then , where , for all (see Figure 4).
Proof. The proof can be obtained directly from the boundedness and height of the poset .
Theorem 6. Let be a bounded poset with , then there is an EA such that .
Proof. Let be the smallest and largest element of a bounded poset , that is: , for any .
Since , by Lemma 4 (see Figure 4). Obviously, is an EA and , where see the follows.
Theorem 7. Let be an EA and . Then the following are equivalent:(1);(2)For all , if is defined then .
Proof. (1) (2). For all , if is defined, then and . In , for all . Thus we have . (2) (1). For all , when , we have or . Then , such that and by (2).Since by Lemma 3, for all . Hence , the proof is complete.
Theorem 8. Let be an EA with , . Thenwhere , .
Proof. For all , andIt is easy to verify that is an EA andfor all . Thus, is defined and , therefore, is defined iff by Theorem 7. Hence , the proof is complete.
Corollary 2. Let . In the isomorphism sense, there are altogether different homo-ordered effect algebras with as the induced partial ordered set.
Remark 6. (1)In Theorem 8, when , . Therefore, the effect algebra with is obtained by some 2-element effect algebras and 3-element effect algebras through and operations.(2)We find out the structure of the EA of height 2 of its partial ordered set.
The structure of the EA of height 3 of its partial ordered set. Here are some examples.(1) of :(2) of :
Example 6. , and are effect algebras whose partial ordered sets have height 3. But the poset of is not a lattice, and , is a cube (see Figure 5(a)).
(a)
(b)
Here is another example of an EA whose poset is not a lattice.
Example 7. It is easy to verify that is an EA, where see the follows.
is not a lattice (see Figure 5(b)).
Example 8. The poset in Figure 6is not an induced poset of any lattice effect algebra. At the same time, we notice that at,is all lattice, and we callcrown lattice.
Figure 7 below shows the crown lattice , and with , and 4.
4. Chain Effect Algebra (CEA)
In the previous section we obtained the complete structure of a class of effect algebras.They are constructed from 2-element and 3-element effect algebra by and operations. Since both 2-element and 3-element effect algebras are chain effect algebras, we will discuss chain effect algebras in this section.
Lemma 5. Let be an EA and . Then(1)if then ;(2)if then ;(3) iff there exists a atomsuch that;(4)if then .
Proof. (1)Let , then by Lemma 1 (4). Thus , that is .(2)Since , Hence by (1).(3)If , then , . Let , since , we have is defined and by (1), then and by (2). Hence is atom of . Conversely, let be an atom of , and , then . If there . that is for some . Then , Since is an atom, we have , i.e. . Hence .(4)Since , then , and is atom of by (3). We have by Lemma 1 (4), Then , the proof is complete.
Theorem 9. If is an -element chain , there is and only one effect algebra constructed by poset , and itsandoperations are as follows:
Proof. Obviously, the and operations given in the theorem satisfy the condition (E1) – (E4), that is, is an EA. The order relation induced on is .
The following shows that the effect algebra constructed by is unique.
Let be an EA and the induced poset is the . Since , we haveSince , then , , then by Lemma 5 (1). HenceTherefore, , that isSimilarly, we can show thatConsidering the above mentioned, we can get: . And, according to the above equation, , hence , , i.e. .
Thus, the effect algebra constructed by poset is unique, the proof is complete.
Definition 9. (see[9]). Letbe a partial ordered set.(1)has the ascending chain condition ( ACC ) if it has no infinite strictly ascending sequences, that is, for any ascending sequence , for all .(2)has the descending chain condition ( DCC ) if it has no infinite strictly descending sequences, that is, for any descending sequence , for all .(3)An effect algebra has the ACC ( DCC ) ifhas the ACC ( DCC ). where is induced order of .
Definition 10. (see[9]). A posetis said to have a maximal condition if each non-empty subset of contains a maximal element. Dually, the posetcan be defined to have minimal conditions .
Lemma 6 (see[9]). Let be a poset, then(1)The sufficient and necessary condition for to satisfy ACC is that has the maximum condition.(2)The sufficient and necessary condition for to satisfy DCC is that has the minimal condition.
Theorem 10. Let be an EA, then has the ACC iff it has the DCC.
Proof. If has the ACC and let be descending sequence, i.e.Then therefor, , for all by ACC. Thus for all , and has the DCC, i.e. ACC DCC and vice versa. The proof is complete.
Using the above two theorems, we get the following result.
Theorem 11. A chain effect algebra must is one of the following:(1)is a finite setand(2)have an infinite strictly ascending chain and an infinite strictly descending chain
Proof. If is a finite set. Obviously, (1) is true.
Now let’s assume that C is an infinite set, and let’s prove that can only be (2). In face, fails to have the DCC and ACC (if not, has the ACC, then has the DCC by Theorem 10, hence is a finite set. This is a contradiction.). Hence have an infinite strictly ascending chainObviously,is an infinite strictly descending chain in . The proof is complete.
Here is the simplest example of an infinite chain effect algebra.
Example 9. Let , and define and as follows:Then is an infinite CEA. And
Theorem 12. Let be an EA, its induced order, . If and is defined then .
Proof. Since is defined, , we have is defined and by Lemma 1 (2) and (5).
Let .
For every Thus, .
Corollary 3. Let be a CEA, . Ifis defined then we have:
Theorem 13. Let be an EA with has no atoms. If then such that .
Proof. Consider . So such that by Definition 1 (E1). Since has no atoms, we have: such that , therefore , the result holds.
Corollary 4. Let be a finite EA. If has a atom , such that , , then is a chain.
Proof. For the sequence in , since is finite, we have: , but is undefined.
Since , we have for some . If , then and is defined by Lemma 1 (2). This is a contradiction, hence . Thus . Now let’s drove that is equal to .
Assume that and . Since and , then , , but . HenceObvious, , and , thus . This is a contradiction. Henceand is a chain.
The following example shows that Corollary 4 fails when is an infinite EA.
Example 10. Let andthen is an EA, but is not chain (See Figure 8). In face, is not even a lattice ( has no least upper bound in ).
Example 11. Letthen is an EA, and is a chain (See Figure 9).
We naturally ask the question: in Corollary 4, if is a LEA must be a chain?
The following theorem answers this question.
Theorem 14. Let be a LEA. If has a atom , such that , , then is a chain.
Proof. If such that but is undefined. Then by Corollary 4, we know that the theorem is true. The theorem will be proved in the case where is defined .
Obvious, for all , . According to the proof of Corollary 4, similarly, we can getand its dualSince , we have .
Next, we will prove that is a chain. SinceAssume that are incomparable. Thenand, , . Since is a lattice, we have , let . Since , we have such that . Thenhence , This is a contradiction, thus is a chain.
Definition 11. Let be a lattice, ,(1) is join-irreducible if, ().(2) is meet-irreducible if, (). is join-irreducible} and is meet-irreducible}.
Theorem 15. Let be a LEA. Then the following conditions are equivalent:(1)is a chain.(2)is join-irreducible element of.(3)is meet-irreducible element of.
Proof. (1) (2): Let . Since is a chain, we have: and are comparable. Hence or , that is. or . Thus, 1 is join-irreducible. (2) (3): By Lemma 2. (3) (1): Let be a LEA. Suppose that is not a chain. then have holds.Since is a lattice, , let . Since , we have: , then such that . Hencethat is. , That contradicts the fact that 0 is meet-irreducible element. Thus is a chain.
Corollary 5. Let be a LEA. If 0 is meet-irreducible element of , then .
5. Conclusion
The main content of this paper is to study the properties and structures of LEAs from the perspective of partial ordered sets. We study the characterization of original effect algebras by partial ordered sets induced by EAs. The structure and number of effect algebras generated by bounded partially ordered sets of height 2 are solved.
We study the chain effect algebra and give some necessary and sufficient conditions for determining the LEA as a CEA. It is proved that a finite EA is a CEA if and only if it has only one atom, and some counterexamples are given.
Data Availability
All data from this study are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant nos. 19801016, 10261003).